171sp05_midterm1 - 550.171 Discrete Mathematics - Spring...

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Unformatted text preview: 550.171 Discrete Mathematics - Spring 2005 Name Midterm #1 1. (a) [5 pt] State a careful de¯nition of what it means for a positive integer to be composite. (b) [10 pt] Recall from the ¯rst day of class that if p is a prime such that 2p ¡ 1 is also prime, then we set Mp = 2p ¡ 1 and call Mp a Mersenne prime. Write a careful proof of the following fact (style and grammar counts here): 2 If Mp is a Mersenne prime, then Mp ¡ 1 is composite. 2. (a) [8 pt] How many distinct anagrams of the word ELECTRICITY are there (including nonsensical words)? (b) [7 pt] How many distinct two word combinations can be made from this same word? For example, two such combinations are TREEL CICITY and L TEECITCIYR. 3. De¯ne R = f(a; b) 2 Z £ Z : a and b are both even or both oddg: Then R is a relation on Z. (a) [10 pt] Of the ¯ve properties relations can have, name all the properties R has. No proofs are necessary here. (b) [3 pt] Is R an equivalence relation? YES or NO (circle one.) (c) [7 pt] If your answer to part (b) is YES, then describe [0] and determine the number of distinct equivalence classes. If your answer to part (b) is NO, provide a counterexample. 4. [15 pt] Let a; b; d; x; and y be integers. Prove: if dja and dj b, then dj ax + by. (Warning: clarity, correctness, style and grammar are all crucial here!) 5. [3 pt] What form of proof is required to disprove a statement? (b) [12 pt] Let A; B and C be sets. Disprove: A ¡ (B \ C ) µ A ¡ (B [ C ). Please draw a Venn diagram for this situation as well. 6. [4 pt each] Correctly label each of the following as either TRUE or FALSE. Spell out the word. (a) If A is a set with j Aj = 10, then 290 relations on A are irre°exive. (b) If f2g 2 B , then ff2gg µ B and ff2gg 2 2B. (c) x ! y is logically equivalent to (x ^ :y) ! FALSE. (d) ff1g; f2; 3g; f4g; ;g is a partition of f1; 2; 3; 4g. (e) The number of lists of length n (here n ¸ 1 is an integer) that can be formed by elements of the set B = fA; C; T ; Gg, where elements from the set B may be repeated is n4. BONUS. Compute 1 1 1 1 1 1 1 1 1 + + + + + + + + : 8!0! 7!1! 6!2! 5!3! 4!4! 3!5! 2!6! 1!7! 0!8! Leave your answer as the ratio of two integers. ...
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This note was uploaded on 03/19/2010 for the course EOE 342 taught by Professor Jooe during the Spring '10 term at Albany College of Pharmacy and Health Sciences.

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